TUTORIAL KIT
OMEGA SEMESTER
PROGRAMME: BANKING AND
FINANCE
COURSE: BFN 425
QUANTITATIVE TECHNIQUE FOR FINANCIAL DECISIONS
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DISCLAIMER
The contents of this document are intended for practice and leaning purposes at the
undergraduate level. The materials are from different sources including the internet
and the contributors do not in any way claim authorship or ownership of them. The
materials are also not to be used for any commercial purpose.
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BFN 425: QUANTITATIVE TECHNIQUE FOR FINANCIAL
DECISIONS
CONTRIBUTORS: Dr A. E. Omankhanlen, Dr. F. O. Olokoyo and Mr. Areghan Isibor
TUTORIAL QUESTIONS
1. Highlight five phases of quantitative analysis and explain briefly.
2. Model is a simplified representation of reality. Discuss?
3. Explain five types of quantitative models that can be employed in making financial
decisions.
4. List and explain the two classifications of a model.
5. Quantitative models are of no relevance to financial decision making”. Discuss and
justifyyour position on this statement.
6. Explain the various reasons why quantitative approach might be useful in decision making
process?
7. Express Commercial Bank Plc provides loans and advances to its customers within the
available savings deposits (SD) and time deposits (TD) available to it. Each advance (A)
absorbs 6 SD and 4TD and each loan (L) takes ISD and 2TD. The bank has only 8SD and
16TD at its disposal. Each overdraft contributes #8.00 profit and loan #10.00 profit.
Required:
(i). Formulate a linear programming model for the bank to maximize the profit by allocating
SD and TD between overdraft and loans.
(ii). Using the simple algorithm, find an optimum solution to the problem in (a).
(iii). If each of the constraints is increased by I unit, what will be the effect on profit?
8.Differentiate between Normative and Descriptive Models.
9. Odua Electric Company produces two products P1 and P2. The products are produced and
sold on a weekly basis. The weekly production cannot exceed 25 for product P1 and 35 for
product P2 because of limited available facilities. The company employs total of 60 workers.
Product P1 requires two man-week of labour while P2 requires one man-week of labour.
Profit margin on P1 is #60 and on P2 is #40.
Required:
Formulate this as a LP problem and solve for maximum profit graphically.
10.Explain the various phases of quantitative analysis?
11. Discuss the nature and relevance of Decision Theory to financial decision making.
12.There are some MP models under conditions of certainty and some under condition of
risk and uncertainty. Discuss and show their classification
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13. Assume Dr. Alex Ajayi has just bought a second hand deep-freezer from Dr. Roland
Ikpefan to sell a new brand of iced tea called Icex in Canaanland. The selling price of icex is
assumed to be N6000 and the cost price is N4800. Assume also that the daily demand for icex
are 26,28,30 and 32 cartons with associated probabilities of 0.2, 0.4, 0.1, and 0.3 respectively.
Before the purchase of the deep-freezer, Dr. Roland has been buying 28 cartons of icex for
sale every morning, but Dr. Alex is not certain whether it is a good policy or that he should
continue with the act.
Required:
As a quantitative financial expert, you are requested to advise Dr. Alex on the best policy of
the appropriate number of cartons to be purchased each morning in order to make
maximum returns, following these steps:
i.
Construct the payoff table for the transaction
ii.
Determine the highest expected payoff
iii.
Recommend the best act that will maximize his returns
14. Explain any four features of LP
15. Explain Games Theory. Discuss five assumptions of games theory.
16. Explain any three advantages and disadvantages of LP
17. Consider the application of games theory to resolve the conflict situation between the
NLC and the Federal Government over a new tax relief package (TRP), in which the
motives of the two players are dichotomized. Each of the two participants has four major
strategies to play the game, viz:
I – Threat and Strike
II – Lobbying and Pressurising Action
III – Arbitration and Court Action
IV – Tolerance and Reconciliation Action
The costs to the Federal Government are given for every pair of strategy choice as follows:
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Player B
(Federal Government)
Strategies
Strike and
Lobbying and
Arbitration
Tolerance and
Threat
Pressuring
and Court
Reconciliation
I
II
III
IV
I
20
15
12
35
II
25
14
8
10
III
40
2
10
5
IV
-5
4
11
0
Strike and Threat
Player A
(NLC)
Lobbying and
Pressuring
Arbitration and
Court
Tolerance and
Reconciliation
Required:
i.
Find the value of the game.
ii.
Determine the best strategy for NLC
iii.
Determine the best strategy for the federal government
iv.
State the implications of two-personszero-sum in the game
v.
State two merits and two demerits of this game.
18.Define games theory and explain any five of its assumptions
19. A company is considering investing in three kinds of projects. The company has an
overriding objective to obtain a return of 15% under condition of minimum risk. The
expected return for the first project is 8% while the second and third projects have expected
returns of 12% and 20% respectively. Project one has standard deviation of its return as
0.02, project two as 0.0245 and project three as 0.0283. The company has a total of N1m to
invest on the projects and it is assumed that the returns on the projects are statistically
independent.
Required: Analyse and explain with the appropriate mathematical algorithm, how the
company can minimize the risks in the face of the constraints.
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20. Explain any four of the following concepts used in games analysis:
A) Strategy B) Value of the game C) Payoff Matrix D) Optimal strategy E) Two person zerosum game.
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ANSWERS
Question 1
i. Formulation of the problem: this is the first step in conducting quantitative analysis. it involves
the identification and formulation of the actual problem to be investigated. The problem to be
investigated must be clearly specified so that correct answer could be provided. The main objective
of the study must be described to reflect an accurate description of the overall interest of the
system. The system will be saved from producing inconsistent and unacceptable results, if the
decision alternatives and constraints are taking into consideration.
ii. Specification of a model: the model should be a quantitative description of the objectives and
constraints of the problem in terms of its decision variables. This type of model makes it possible
for the operation research team to analyze the system and examine various alternatives without
disrupting the behaviour of the entire system. As a framework of analysis, the model also helps to
show what variables are relevant and what data are needed for the description of the system.
iii. Estimation of the model: this is the third phase of quantitative analysis. It involves model
estimation for the purpose of providing accurate solutions to the problem. This involves fixing of
data collected as input into the model. It is at this stage that the model is tested. Testing of a
quantitative model may involve application of a well-defined optimization techniques through an
optimum solution to the model may be determined. However since a mathematical model is a mere
representation of the real problem, the resulting optimum solution, may not guarantee the best
solution to the real system. But a good approximation to the optimum solution may suffice. In
addition, the behaviour of the solution can further beanalysed by what is called sensitivity analysis.
It explains what happens to the solution or how it will behave if deliberate changes are made in
the system's parameter.
iv. Validation of the model: the next stage after securing the solution is to validate both the model
and the solution obtained. A model may be adjudged valid if in the face of its limiting assumption
and constraint, it gives a reliable or accurate prediction of the performance system. There are two
common approaches which are relevant for assessing the validity of a model. One approach is by
comparing the model's performance with the actual performance on the basis of past data. The
second is to compare the model's performance with the behaviour of the system under the condition
of no change. The predictive ability of the model can therefore be ascertained on the basis of these
two approaches.
v. Implementation of output: this is the last stage of quantitative analysis and it involves the
implementation of the results obtained from the model tested. This requires that the analyst should
translate the model's output in unambiguous manner to operating procedure for the understanding
of the party responsible for the operation of the system. After the implementation of the output,
the response of the system to the changes should be objective and necessary adjustments made if
need be. It suggests that there should be cooperation between operation research analyst and the
party involved in operating and managing the system.
Question 3
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i. Iconic model: They are visual models of real life situations or problems in form of symbols,
pictures objects, diagrams etc. Iconic model is a look alike representation of some specific entity.
ii. Analogue model: This model is used to simplify reality through the use of analogy. Analogue
model uses one type of behaviour of variable or properties to represent another type. It represents
the relationship between two variables by a line on a chart and as a result conveys the relationship
between the variables in an efficient and convenient way. Graphs are the most frequently used
analogue models.
iii. Simulation model: Simulation model consist of a series of equations and formula arranged
systematically so that they "behave" in a similar manner to the real system being investigated. This
type of model is relevant where there are no appropriate mathematical models to capture a problem.
iv. Mathematical or symbolic models: they explain the behaviour of the real world by means of
mathematical and statistical notation, symbols, equations and formulae. Symbolic models are often
used whenever the reality is too complex or too abstract to be portrayed through an iconic model
or analogue model
v. Heuristic Model: refers to techniques based on experience for various tasks such as research,
problem solving, discovery and learning. These models attempt to show how to arrive at a
workable solution to a problem or a better option to a current practice by following a set of
intuitive rules or direction.
Question 5
Quantitative models are relevant to financial decision making because it can be applied in the
following areas of financial operations:
i. Cash management: Quantitative technique like linear programming, can be used to achieve the
objective by determining the proportion of total funds that should be allocated to the various units,
given the available constraints.
ii. Capital budgeting: Quantitative techniques can be used in the process of investing an
organization’s current fund most efficiently in long term activities in anticipation of an expected
flow of future benefits over a series of years.
iii. Inventory control: The Economic Order Quantity (EOQ) technique used in quantitative
analysis for inventory management can help the financial managers to develop optimal inventory
policies and reduce the investment in inventories or their waste.
iv. Determination of an organization’s optimal capital structure: It is also useful in determining
an organization’s optimal capital structure. Capital structure involves the analysis of the
combination of a company's selected bonds, debentures, preferred stocks and common stock.
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v. Investment Analysis and Portfolio Management: Quantitative models can also be applied in
portfolio management in order to get an optimum or efficient portfolio.
Question 7
(i) State the constraints and objective functions. The portfolio must satisfy the inequalities.
Let:
X1 = Advance
X2 = Loan
Constraint Function:
6X1 + 1X2 ≤ 8 (SD Constraint)
4X1 + 2X2 ≤ 16 (TD Constraint)
Objective Function:
Ⱬ = 8X1 + 10X2
(Profit Objective Function)
Hence: Maximize the objective: Ⱬ = 8X1 + 10X2
Subject to the constraints: 6X1 + 1X2 ≤ 8
4X1 + 2X2 ≤ 16
State the constraints in standard form by turning inequalities into equations.
6X1 + 1X2 + 1S1 + 0S2 + 0z = 8
4X1 + 2X2 + 0S1 + 1S2 + 0z = 16
8X1 + -10X2 + 0S1 + 0S2 + 1z =0
State the coefficient of variables (values):
8= 6 1 1 0 0
16 = 4 2 0 1 0
0 = -8 -10 0 0 1
This is presented in the first simplex tableau, showing coefficient and variables.
Tableau 1
leave
Row no Variables Values X1
SD
1
S1
8
6
TD
2
S2
16
4
Index
3
Ⱬ
0
-8
enter ●
X2
S1
1
1
2
0
-10
0
●
S2
0
1
0
Ⱬ How much
0 8/1 = 8
0 16/2 =8
1
The Pivot Column = X2
The Pivot Row = TD
The Pivot Coefficient = 2
Next: Fill the rows in a second table by using the following formulas:
New 2ndRow =Old Pivot Row = 16 4 2 0 1 0
Pivot Coefficient
2
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(8 2 1 0 0.5 0)
New 1st Row= Old first row – 1 (new second row)
= 8 6 1 1 0 0 - 1 (8 2 1 0 0.5 0)
= 8 – 8, 6 – 2, 1 – 1, 1 – 0, 0 - 0.5, 0 – 0
=0
4
0 1 -0.5 0
New 3rd Row = Old index row – (-10) (new second row)
= 0 -8 -10 0 0 1 – (-10) (8 2 1 0 0.5 0)
= 0 -8 -10 0 0 1 + (80 20 10 0 5 0)
= 0 + 80, -8 + 20, -10 + 10, 0 + 0, 0 + 5, 1+ 0
= 80
12
0
0
5
1
Table 2
enter ● ●
Row no Variables Values X1
X2 S1 S2
Advance
1
S1
0
4
0 1 -0.5
Loans
2
X2
8
2
1 0 0.5
Index
3
Ⱬ
80 12
0 0 5
Ⱬ How much
0
0
1
(ii). The optimum solution can be obtained if the bank:
 N0m on advance
 N8m on Loans and
 The expected profit is N80m
(iii). If loans constraint is increased by 1 unit, the profit will also increase by its shadow price of
N5m. If advance constraint is increased by 1 unit, there will be no increase in profit i.e N0m
shadow price as shown in the index row of table 2.
Question 9
Step 1: Define the decision variables
Let X1 = No. of units of P1 to be produced
X2 = No. of units of P2 to be produced
Maximize the total Naira contribution.
Ⱬ = N60X1 + N40X2
Subject to the constraint equations:
Production capacity: 2X1 + 1X2 ≤ 60 hours
Minimum Requirement: X1 ≤ 25 units
X2 ≤ 35 units
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Step 2: The solution to the formulated problems can be obtained graphically as follows:
60
X1
50
Minimum Requirement X2≤ 35
40
30
A
B
Minimum Requirement X1≤ 25
Feasible Region
20
C
Production capacity 0.05X1 + 0.1X2≤ 80hrs
10
0
10
20
30
40
Production Capacity: X1 = 60/2 = 30
X2 =60/1 = 60
50
60
X2
Step 3: The result of the solution can be shown in a table as follows:
Production points
X1 (Units)
X2 (Units)
A
B
C
25
25
15
10
35
35
Value of the obj
function (Naira)
1900
2900
2300
Solving for the value of the objective function we will use Ⱬ = N60X1 + N40X2
Point
A
B
=
=
X1
N 60 x 25
N 1200
N 60 x 25
N 800
+
+
X2
N 40 x 10
N1200
= N 190
+
+
N 40 x 35
N 1800
= N 2900
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C
=
N 60 x 15
N 1400
+
+
N 40 x 35
N 1200
= N 2300
Decision: The objective function is maximized at point B (25 units of P1 and 35units of P2) where
it equals N 2900. Hence, the firm should produce 25 units of product P1 and 35units of P2.
Question 11
Nature and Relevance of Decision Theory to Financial Decision Making:
i.
Decision analysis refers to application of quantitative data and rational
approach to problem of decision making under uncertainty.
ii.
It provides the financial manager with the appropriate tools for determination of
optimum decisions in the face of uncertainty.
iii.
It has wide applicability and relevance to the field of finance and other studies
Question 13
i. The pay-off table is constructed as follows:
13 packs
14 packs
15 packs
16 packs
Outcome Demand Prob Sales PayProb Sales PayProb
Sales PayProb
Sales PayProb
off
x
off
x
off
x
off
x
paypaypaypayoff
off
off
off
X1
26
0.2
26
31200 6240 26
21600 4320 26
12000 2400 26
3000 600
X2
28
0.4
26
31200 12480 28
33600 13440 28
24000 9600 28
15000 6000
X3
30
0.1
26
31200 3120 28
33600 3360 30
36000 3600 30
27000 2700
X4
32
0.3
26
31200 9360 28
33600 10080 30
36000 10800 32
39000 11700
EXPECTED PAY-OFF
30840
31200
26400
21000
ii. The highest expected pay-off is N31200 which corresponds to the act of purchasing 28 cartons
every morning.
iii. It is recommended that Dr. Alex Ajayi should purchase 28 packs of ‘icex’ for sale every
morning.
Question 15
Games theory deals with analysis of decision making when two or more intelligent and rational
opponents are involved under conditions of conflict and competition.
Assumptions of Game Theory:
 Each player aims at maximising gains and minimizing losses.
 Each player has a definite set of probable courses of action or strategies available to him.
 Each player has the knowledge of both strategies
 Each player behaves naturally
 The first player’s gains are the second player’s losses and vice versa.
Question 17
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Player B
(Federal Government)
Strike and
Lobbying
Arbitration
Tolerance
Row
Threat
and
and Court
and
minim
Reconcilia
um
Pressuring
Strategies
Maximin
tion
Player
A
I
II
III
IV
20
15
12
35
12
25
14
8
10
8
Strike and Threat
(NLC)
I
Lobbying and
Pressuring
II
12
Arbitration and
Court
III
40
2
10
5
2
IV
-5
4
11
0
-5
40
15
12
Tolerance and
Reconciliation
Col. maximum
Minimax
35
12
Saddle Point exists: maximin = minimax = 12
i.
The value of the game = 12
ii. The best strategy for NLC is strategy I, Strike and Threat.
iii. The best strategy for Federal Government is strategy III which is Arbitration and Court.
iv.
The implication of two-persons-zero sum game is that the gains of NLC in the game are
the losses of the Federal Government.
Question 19
Non-Linear Mathematical Programming Model
Let X1, X2, X3 represent projects 1, 2, 3 respectively.
Total Fund Constraint
X1+X2+X3 = # 1000,000
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X1+X2+X3 = 1000
The Expected returns Constraint
0.08X1 + 0.12X2 + 0.2X3 = 0.15 (1000)
0.08X1 + 0.12X2 + 0.2X3 = 150
The risks of the Portfolio (Objective Function)
The risks is measured by the variance i.e. (S.D) 2
Note:The objective is to minimize the risk.
Z = (0.02X1)2 + (0.0245X2)2 + (0.0283X3)2
= 0.0004X12 + 0.0006X22 + 0.0008X32
Step 1: Modify the Constraints:
From the first constraint we obtain
X1 = 1000 – X2 –X3………………………………………… (1)
Step 2: Substitute equation (1)for (X1) in the Second Constraint
0.08(1000 – X2 – X3) + 0.12(X2) + 0.2(X3) = 150
80 – 0.08X2 – 0.08X3 + 0.12X2 + 0.2X3 =150
0.12X2 – 0.08X2 + 0.2X3 – 0.08X3 =150 - 80
0.04X2 + 0.12X3 =70
0.04X2 = 70 – 0.12X3
X2 = 70 – 0.12X3
0.04
X2 = 1750 - 3X3…………………………………………(2)
Step 3: Substitute equation (2) for X2in equation (1)
X1 =1000 – X2 – X3
X1 = 1000 – (1750 -3X3) – X3
X1 = -750 +2X3
X1 = 2X3 -750
Step 4: Substitute the values for X1 and X2 into the objective function
Recall that Z = 0.0004X12 + 0.0006X22 + 0.0008X32
= 0.0004 (2X3 -750)2 + 0.0006 (1750-3X3)2 + 0.0008X32
=0.0004 (4X32- 3000X3 +562,500) +0.0006 (9X32- 10500X3+ 3,062,500) + 0.0008X32
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= 0.0016X32- 1.2X3 +225 + 0.0054X32 – 6.3X3 + 1838 +0.0008X32
Collect like terms;
Z = 0.0016X32 + 0.0054X32 + 0.0008X32 – 1.2X3 – 6.3X3 + 225 + 1838
= 0.0078X32 – 7.5X3 + 2063
(Quadratic problem)
Step 5: Solvefor X3 using the Quadratic Formula
Z =0.0078X32 – 7.5X3 + 2063
a = 0.0078, b= -7.5 and c = 2063
-b + b2 – 4ac
2a
7.5 +
(-7.5)2 – 4 (0.0078) x 2063
2 x 0.0078
7.5 + 56.25 -64.36_____
0.0156
7.5 + -8.12____
0.0156
7.5+ 0
0.0156
X3 = 480.77
X2 = 1750 – 3X3
= 1750 - 3(480.77)
= 1750 – 1442.31
= 307.69
X1 = 2X3 - 750
= 2 (480.77) – 750
= 211.54
X1 = #211,540, X2 = #307,690 and X3 = #480,770
Expected Return
Stock 1 = 0.08X1 = 0.08 (211,540) = 16923
Stock 2 = 0.12X2 = 0.12 (307,690) = 39923
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Stock 3 = 0.12X3 = 0.12 (480,770) = 96154
#150,000_
DECISION:
The investor should invest in #211,540 project 1, #307,690 in project 2 and #480,770 in project3
to be able to achieve the expected return of 15% on the #1,000,000 = #150,000
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